Question: Solve for $x$ : $ 5|x + 3| + 2 = 2|x + 3| + 8 $
Answer: Subtract $ {2|x + 3|} $ from both sides: $ \begin{eqnarray} 5|x + 3| + 2 &=& 2|x + 3| + 8 \\ \\ { - 2|x + 3|} && { - 2|x + 3|} \\ \\ 3|x + 3| + 2 &=& 8 \end{eqnarray} $ Subtract ${2}$ from both sides: $ \begin{eqnarray} 3|x + 3| + 2 &=& 8 \\ \\ { - 2} &=& { - 2} \\ \\ 3|x + 3| &=& 6 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x + 3|} {{3}} = \dfrac{6} {{3}} $ Simplify: $ |x + 3| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 3 = -2 $ or $ x + 3 = 2 $ Solve for the solution where $x + 3$ is negative: $ x + 3 = -2 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& -2 \\ \\ {- 3} && {- 3} \\ \\ x &=& -2 - 3 \end{eqnarray} $ $ x = -5 $ Then calculate the solution where $x + 3$ is positive: $ x + 3 = 2 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& 2 \\ \\ {- 3} && {- 3} \\ \\ x &=& 2 - 3 \end{eqnarray} $ $ x = -1 $ Thus, the correct answer is $x = -5 $ or $x = -1 $.